Bayesian inference to identify material parameters in solid mechanics

Bayesian inference to identify material parameters in solid mechanics

A commonly employed approach to identify material parameters in mechanics is the standard least squares method in which the sum of the squared residuals (i.e. the difference between the measurement and the model response) is minimised with respect to the parameters of interest. An alternative way is the Bayesian approach which can provide a quantification of uncertainty. Bayesian inference results in a probability density function (PDF), a so-called posterior distribution, which is a function of the material parameters of interest and represents our uncertainty about the material parameter. Once the posterior is established, the statistical summaries of the material parameters, e.g. the mean values of material parameters, the material properties at which the PDF is maximum (called the ‘maximum-a-posteriori-probability’ or ‘MAP’ point) and the standard deviation (i.e. a quantity that shows how much measurements for a group are different from their mean) can be obtained by analysing the posterior distribution. This analysis can be either by computing the statistical moments of the posterior (e.g. the first moment of a PDF is its mean value) or by simulating samples from the posterior. It is important to note that from a Bayesian perspective, probability distributions are not real things but are models chosen, or derived from modelling assumptions, to represent the user’s uncertainty about the values of quantities such as model parameters and measurement and model prediction errors.In this framework the prior knowledge (so-called prior distribution which chosen by the user) is updated by data through Bayes’ theorem.

Schematic of updating our original belief (prior) based on new observations using BI.

The prior, the posterior and the value predicted by least squares method.

Aims

Two major goals are defined in this project: (1) employing Bayesian inference for parameter identification in the solid mechanics and (2) developing a scheme that enables us to identify material parameters joint PDF (i.e. the PDFs that represents the material randomness) with limited number of experiments/experimental specimens.

The second case can be used to identify material randomness in structures of discrete fibres, yarns or struts. Material randomness can be considered as one of the factors that can characterise the structural behaviours of these structures at a small length scale. One possible way to identify the material randomness is to test numerous small fibres, yarns of struts but this entails a substantial amount of experimental efforts. In our study, we employ Bayes’ theorem to only test a relatively small number of specimens and use their results to infer the parameters of an initially assumed distribution.

True joint PDF from which we have assumed the material parameters are drawn. The specimens are considered to behave base on the linear elastic-perfect plastic law.

Estimated joint PDF. The scheme that has been employed to identify the parameters of the joint PDF is based on modular BI.